## Exponential Convergence of hp-FEM
for Maxwell's Equations with Weighted Regularization
in Polygonal Domains

*
Martin Costabel,
Monique Dauge,
Christoph Schwab*

The time-harmonic Maxwell equations
do not have an elliptic nature by themselves.
Their regularization by a divergence term is a standard tool
to obtain equivalent elliptic problems.
Nodal finite element discretizations of Maxwell's equations obtained
from such a regularization converge to wrong solutions
in any non-convex polygon.

Modification of the regularization term consisting in the
introduction of a weight restores the convergence of nodal FEM,
providing optimal convergence rates for the **h Version**
of Finite Elements, see **Paper**.

We prove exponential convergence of **hp FEM** for the
weighted regularization of Maxwell's equations in plane polygonal
domains provided the hp-FE spaces satisfy a series of axioms.
We verify these axioms for several specific families
of hp finite element spaces.

16 June 2004.

SAM Report, *ETH Zürich,* N. ** 2004-05 **.

M^{3}AS: * Mathematical Models and Methods in Applied Sciences,*
** Vol. 15, n° 4, 2005**, 575-622.